# Persistent-Noise Multi-View Fusion

- ID: 142
- Canonical URL: https://synthcore.org/idea/142/persistent-noise-multi-view-fusion
- API JSON: https://synthcore.org/api/idea/142.json
- API Markdown: https://synthcore.org/api/idea/142.md
- Verification status: audited
- Source: [arXiv:2608.24819](https://arxiv.org/abs/2608.24819)
- Category: training
- Solves: accuracy, stability, generalization
- ML areas: loss, training, regularization
- Math tags: probability, information-theory, statistics
- Ratings: usefulness 6/10; difficulty 4/10; novelty 7/10

## Idea description

Train a classifier or encoder to distinguish shared latent corruption from fresh per-view noise instead of treating repeated observations as conditionally independent given the target. A single persistent state corrupts all views, while each view may additionally receive independent observation noise; the fusion loss marginalizes the persistent state exactly. This should reduce overconfident predictions from repeated but systematically biased augmentations, sensor readings, or retrieved documents.

## Mathematical statement

The paper defines a finite-alphabet persistent channel with clean symbol X in alphabet \(\mathcal X\), persistent intermediate symbol U in alphabet \(\mathcal U\), and transition matrix T. The key equation is \(T_{xu}(q,R)=(1-q)\mathbf{1}\{u=x\}+qR_{xu}\), where \(q\in[0,1]\) is the probability of a persistent deviation, R is a fixed row-stochastic matrix with \(R_{xx}=0\), and \(\mathbf{1}\{u=x\}\) is one when the persistent state is unchanged. For a neural-network adaptation, let x be the clean class or latent token, u be a shared corruption state sampled once per example, and \(y_{1:K}\) be K observations generated conditionally independently from u through an observation matrix S, where \(S_{uy_k}=p(y_k\mid u)\). The correct likelihood is \(p(y_{1:K}\mid x)=\sum_u T_{xu}\prod_k S_{uy_k}\), not \(\prod_k\sum_u T_{xu}S_{uy_k}\). With prior \(\pi_x=p(x)\), the exact posterior is \(p(x\mid y_{1:K})=\frac{\pi_x\sum_u T_{xu}\prod_k S_{uy_k}}{\sum_{x'}\pi_{x'}\sum_u T_{x'u}\prod_k S_{uy_k}}\). The independent-view expression redraws the persistent state for every observation and can therefore accumulate spurious evidence about a systematic error as K grows.

## Key formulas

- $$T_{xu}(q,R)=(1-q)\mathbf{1}\{u=x\}+qR_{xu},\qquad q\in[0,1],\quad R_{xx}=0,\quad \sum_u R_{xu}=1.$$
- $$p(y_{1:K}\mid x)=\sum_{u\in\mathcal U}T_{xu}(q,R)\prod_{k=1}^{K}S_{uy_k},$$
- $$p(x\mid y_{1:K})=\frac{\pi_x\sum_{u\in\mathcal U}T_{xu}(q,R)\prod_{k=1}^{K}S_{uy_k}}{\sum_{x'\in\mathcal X}\pi_{x'}\sum_{u\in\mathcal U}T_{x'u}(q,R)\prod_{k=1}^{K}S_{uy_k}},$$
- $$p_{\mathrm{ind}}(y_{1:K}\mid x)=\prod_{k=1}^{K}\left(\sum_{u\in\mathcal U}T_{xu}(q,R)S_{uy_k}\right)\neq p(y_{1:K}\mid x).$$

## Implementation notes

Integrate this at the data-loader and fusion-loss level for a small multi-view classifier, or at the token-evidence aggregation level of a transformer. For each training example with clean target x, create one persistent state u using T: with probability 1-q set u=x; otherwise sample u from row x of R. Generate K observations y_k from u using an observation corruption operator S. For images, S can be a class-confusion channel applied after independent image augmentations; for discrete tokens or sensor categories, S can be a lookup-table sampler. Feed all K views through a shared encoder and either concatenate their representations or average their per-view logits. Compute the persistent target distribution with log-sum-exp. For every candidate class x, calculate a_x = logsumexp over u of [log T[x,u] plus the sum over k of log S[u,y_k]], then set target_x = softmax(log pi[x] + a_x). Minimize cross-entropy between the network output and this target, optionally adding the ordinary clean-label loss. As a control, train with the independence likelihood, which multiplies the per-view marginal after effectively redrawing u for every view. Initially set q and R from the synthetic corruption generator; in a second experiment, learn q through a sigmoid parameter and learn R through row-wise softmax, with S fixed. The first cheap experiment is MNIST or CIFAR-10 with a fixed class-confusion matrix R, q in {0.1, 0.3, 0.5}, and K in {1, 2, 4, 8}. Compare ordinary mean-logit fusion, independent-marginal training, and persistent-state marginal training using the same small CNN and augmentations. Measure accuracy, true-label NLL, expected calibration error, and confidence growth with K on systematically corrupted examples. Success means lower NLL and calibration error at equal accuracy, with confidence saturating rather than becoming unjustifiably extreme as K increases.

## Verification

- Status: audited
- Mechanism evidence: yes
- Mechanism confirmed: no
- Verdict: Built exact persistent-state and independent-view fusion with stable log-sum-exp computation, vectorized synthetic sampling, and numerical identity checks. The core formulas matched direct computation with maximum error 3.6e-15, while the persistent model maintained low ECE (0.002-0.009) and NLL 0.868 at K=16 versus independent fusion's ECE 0.347 and NLL 4.774; accuracy was similar (0.652 vs 0.652). This is a clear calibration/NLL win under systematic shared corruption, not an accuracy improvement.

### Mechanism check

- Verdict: Built exact persistent-state and independent-view fusion with stable log-sum-exp computation, vectorized synthetic sampling, and numerical identity checks. The core formulas matched direct computation with maximum error 3.6e-15, while the persistent model maintained low ECE (0.002-0.009) and NLL 0.868 at K=16 versus independent fusion's ECE 0.347 and NLL 4.774; accuracy was similar (0.652 vs 0.652). This is a clear calibration/NLL win under systematic shared corruption, not an accuracy improvement.
- Confidence: 9/10
- Limitations: Only a finite-alphabet synthetic channel was tested with known T and S; no learned CNN/encoder, trainable q or R, image augmentations, FLOP/speed comparison, or real dataset evaluation was included. The independent control is the mathematically misspecified redraw model rather than a neural mean-logit fusion baseline.

## Artifacts

- [experiment.py](https://synthcore.org/code/27/experiment.py)
- [report.md](https://synthcore.org/code/27/report.md)
- [report_super_2026-08-30T102502.md](https://synthcore.org/code/27/report_super_2026-08-30T102502.md)
- [results.json](https://synthcore.org/code/27/results.json)
- [supercheck.py](https://synthcore.org/code/27/supercheck.py)
- [supercheck_report.md](https://synthcore.org/code/27/supercheck_report.md)
- [supercheck_results.json](https://synthcore.org/code/27/supercheck_results.json)
- [Download all files as ZIP](https://synthcore.org/download/27)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.